Lecture 9

Working with Non-Binary Covariates

Today

We’re going to look at the relationship between income and education.
We’ve done that before. When we did, we thought of education in dichotomous terms.
- We had some people with 4-year degrees — the green dots.
- We had some people without them — the red dots.
Today, we’ll be a bit more granular. We’ll think of education in terms of years of schooling.
- 8+ years = finished middle school
- 12+ years = high school diploma
- 16+ years = 4-year college degree

Our Sample

As before, we’ll work with data from the 2022 Current Population Survey.
We’ll look at California residents age 25-35 with at least an 8th-grade education.
This sample includes 2271 people.
- To visualize it, I’ve marked each one on a map.
- To display some of the information we have, I’ve made a table.
I’ve used color to emphasize the dichotomous view of education we’ve been using.

	income	education	age	county
1	$10k	12	28	los angeles
2	$40k	14	27	kern
3	$28k	18	33	sacramento
4	$48k	16	27	san luis obispo
5	$68k	13	33	los angeles
6	$180k	18	33	alameda
7	$60k	16	26	unknown
8	$36k	16	28	los angeles
⋮
2270	$0k	13	32	orange
2271	$47k	13	25	unknown

Visualization

Our primary visualization will be a scatter plot. It shows a dot for each person in our sample.
The person’s income is on the y-axis and their education on the x-axis.
We add a bit of fuzz to the displayed x-coordinate to space the dots out.
- But not so much that it’s not clear what the real value of $X$.
- This is called ‘jittering’. Sometimes a plot like this is called a ‘jitter plot’.
As before, we can use color to highlight the categories of people with and without 4-year degrees.

The Same Thing Without Fuzz

Summarization

Looking at the dots alone, we can get rough answers to some simple questions.
- Which group tends to make more money—people with or without 4-year degrees?
But it’s hard to be precise just eyeballing things. It helps to look at numerical summaries.
I’ve overlaid — in black — the mean income $\pm$ one standard deviation for each group.
- That’s shown by the dot’s position and its ‘arms’ on the y-axis.
- Its position on the x-axis shows the mean years of schooling within the group.

The Dichotomous Version

It’s worth visualizing the information lost by dichotomizing education.
To do this, we can look at a similar plot without that information. This one.
In it, I’ve replaced the x-coordinates with random values that tell us nothing beyond group membership.
People without 4-year degrees get random values uniformly distributed between 8 and 14.
- That’s the range of years of schooling in that group.
People with 4-year degrees get random values between 16 and 20. Same deal.

Fine Comparisons

Let’s think about what we can do with the additional information we have now.
One simple option is to work with the finest grouping we can.
We can, e.g., look at the income mean and standard deviation for people with 8,9,10, etc. years of schooling.
Or compare any pair of those groups. It’s natural to look at groups that are in some sense adjacent.
- 18 vs. 16: the value of a masters’ degree (only) vs. a 4-year one (only).
- 16 vs. 14: the value of a 4-year degree (only) vs. a 2-year degree (only).
- 14 vs. 12: the value of a 2-year degree (only) vs. a high school degree (only).

Fine Comparisons

This is essentially the same as what we did when we dichotomized education.
- We just ‘forget’ the other groups. We don’t look at them.
- That’s great if we know exactly what we’re interested in—and it’s something like that.
But it’s not so great if we’re a bit less focused.
- That’d mean we might have to think about/report a lot of different numbers.
- It can get to be too much. We’re not summarizing enough.

Coarsening

One remedy is to go back to working with coarser groupings.
We can make up own by aggregating—combining—the fine groups we have.
Here, for example, is a breakdown in 4 groups.
- < 14 years: no college degree.
- =14 years: 2-year degree.
- =16 years: 4-year degree.
- > 16 years: graduate degree.

This is more information than a two-group comparison but perhaps little enough to be manageable.
- We can report the set of group-specific summaries.
- Or visualize them in a plot like this one.

Coarsening is Always Happening

                          education=case_match(cps.data$a_hga,
                                           0  ~ 0,   # child
                                           31 ~ 0,   # < grade 1
                                           32 ~ 4,   #  grade 1-4
                                           33 ~ 6,   #  grade 5-6
                                           34 ~ 8,   #  grade 7-8 
                                           35 ~ 9,   #  grade 9
                                           36 ~ 10,  #  grade 10
                                           37 ~ 11,  #  grade 11
                                           38 ~ 11,  #  grade 12 no diploma
                                           39 ~ 12,  #  high school grad
                                           40 ~ 13,  #  some college
                                           41 ~ 14,  #  associate degree (vocational)
                                           42 ~ 14,  #  associate degree (academic)
                                           43 ~ 16,  #  bachelors degree
                                           44 ~ 18,  #  masters degree
                                           45 ~ 20,  #  professional school degree
                                           46 ~ 20)  #  doctorate

Often, the data we have is already coarsened to some degree.
- Our data does not, for example, distinguish between people with 7 and 8 years of schooling.
- Similarly, it doesn’t tell us years for people with graduate degrees. We’ve made that up.
And to fit it into our analysis, we often have to coarsen it further.
- To fit education into our ‘years of schooling’ framework, we’ve grouped together people with …
  - vocational and academic associate degrees. (12+2=14 years)
  - professional and doctoral degrees. (16 + 4ish ≈ 20 years)
- Our choices about some of these made-up numbers create the gaps we see at 15,17,and 19 years.

Keeping all this in mind can be overwhelming.
It’s often a good idea to pick and abstraction — like years of schooling — and go with it.
But it’s important to come back to this at some point to make sure your conclusions mean what you think they do.
Suppose you see a bigger income jump between ‘8’ and 9 years of education than between 9 and 10.
- That might be because the group with ‘8’ years actually includes people with 7 and 8 years.
- So many of the people with ‘8’ years have 2 fewer years of schooling than the people with 9.

Dichotomization

Often, when we want to report a numeric summary, we do resort to dichotomization.
But having the more granular information allows us to choose the groups we want.
There are many meaningful choices.
- >16: people with graduate degrees.
- ≤12: people who haven’t been to college.
Activity. Put these comparisons into words and — for mean income — mathematical notation and R code.
- >16 vs. 16
- 14 vs. ≤12

A Coarsened Comparison: >16 vs. 16

In words

People with graduate degrees vs. people with 4-year degrees (only).

A Coarsened Comparison: >16 vs. 16

In math

Option 1. A compact, readable version \[ \frac{\sum_{i:X_i > 16} Y_i}{\sum_{i:X_i > 16} 1} - \frac{\sum_{i:X_i = 16} Y_i}{\sum_{i:X_i = 16} 1} \]
Option 2. A verbose version that’s easier to use in probability calculations \[ \frac{\sum_{i=1}^n 1_{\{18,20\}}(X_i) Y_i}{\sum_{i=1}^n 1_{\{18, 20\}}(X_i)} - \frac{\sum_{i=1}^n1_{\{16\}}(X_i)Y_i}{\sum_{i=1}^n 1_{\{16\}}(X_i)} \ \ \text{ where } \ \ 1_S(x) = \begin{cases} 1 & \text{ if } x \in S \\ 0 & \text{ otherwise} \end{cases}. \\ \]

A Coarsened Comparison: >16 vs. 16

In code

Option 1. A compact, readable version

mean(Y[X > 16]) - mean(Y[X==16])

[1] 27708.47

Option 2. A verbose version that’s easier to combine with more abstract code

groups = list(a = c(18,20), b=16)
mean(Y[X %in% groups$a]) - mean(Y[X %in% groups$b])

[1] 27708.47

A Coarsened Comparison: 14 vs ≤12

In words

People with 2-year degrees (only) vs. people who haven’t completed a year of college.

A Coarsened Comparison: 14 vs ≤12

In math

Option 1. A compact, readable version \[ \frac{\sum_{i:X_i = 14} Y_i}{\sum_{i:X_i = 14} 1} - \frac{\sum_{i:X_i \le 12} Y_i}{\sum_{i:X_i \le 12} 1} \\ \]
Option 2. A verbose version that’s easier to use in probability calculations

\[ \frac{\sum_{i=1}^n 1_{\{14\}}(X_i) Y_i}{\sum_{i=1}^n 1_{\{14\}}(X_i)} - \frac{\sum_{i=1}^n1_{\{8 \ldots 12\}}(X_i)Y_i}{\sum_{i=1}^n 1_{\{8 \ldots 12\}}(X_i)} \ \ \text{ where } \ \ 1_S(x) = \begin{cases} 1 & \text{ if } x \in S \\ 0 & \text{ otherwise} \end{cases}. \]

A Coarsened Comparison: 14 vs ≤12

In code

Option 1. A compact, readable version

mean(Y[X == 14]) - mean(Y[X<=12])

[1] 12626.81

Option 2. A verbose version that’s easier to combine with more abstract code

groups = list(a = c(14), b=8:12)
mean(Y[X %in% groups$a]) - mean(Y[X %in% groups$b])

[1] 12626.81

The Relationship Between Coarse and Fine Means

We can express the means in our aggregate groups in terms of the means in our fine groups.
Let’s define some notation for describing our fine groups to make this easier.
- $N_x$ will be the number of people with $X=x$.
- $\hat \mu(x)$ will be the mean income of people with $X=x$.

\[ \hat \mu(x) = \frac{1}{N_x}\sum_{i:X_i=x} Y_i \quad \text{ where } \quad N_x = \sum_{i:X_i=x} 1 \]

Q. What, on the plot, corresponds to $\hat\mu(14)$? What about $N_{12}$?

A. The black dot in the $x=14$ column. And the number of red dots in the $x=12$ column.

The Relationship Between Coarse and Fine Means

Q. What, in terms $\mu(x)$, is the mean outcome among people in our sample who did not attend college?
Strategy. Where we see a sum over the people in the aggregate group, we rewrite it as
- a sum over the fine groups
- of sums over the people in them.

Answer. It’s a weighted average of the fine-group means $\mu(x)$. \[ \begin{aligned} \frac{\sum_{i:X_i \in 8 \ldots 12} Y_i}{\sum_{i:X_i \in 8 \ldots 12} 1} &= \frac{\sum_{x \in 8 \ldots 12}\sum_{i:X_i=x} Y_i}{\sum_{x \in 8 \ldots 12}\sum_{i:X_i=x} 1} = \frac{\sum_{x \in 8 \ldots 12} N_x \times N_x^{-1}\sum_{i:X_i=x} Y_i}{\sum_{x \in 8 \ldots 12}N_x} \\ &= \sum_{x \in 8 \ldots 12} P_x \times \hat \mu(x) \quad \text{ for } \quad P_x = \frac{N_x}{\sum_{x \in 8 \ldots 12}N_x}. \end{aligned} \]
The weight $P_x$ is the proportion of the people in the aggregate group with $X_i=x$.

Aggregating Comparisons

Another option—rather than comparing aggregates—is aggregating comparisons.
We can summarize the ‘value’ of each year of high school by the associated increment in mean income.
- Visually, that’s the slope of the dashed black line leading up to it.
To reduce these to a single number, we can take the average of the increments.
- Visually, that’s the slope of the secant—the blue one. In a sense.
- If that’s not obvious to you, don’t worry. We’ll come back to it in a minute.
Why in a sense? Because there are two things we might mean by ‘the average of the increments’.

Different Ways of Aggregating

The average, over the four years of high school, of the increments. Let’s write it out.

\[ \frac{1}{4}\sum_{x=9}^{12} \qty{ \hat\mu(x) - \hat\mu(x-1) } \]

That’s the slope of the secant from $\hat\mu(8)$ to $\hat \mu(12)$—the blue line.
- To see this, we can write out our four terms and see what cancels.
- It helps to count backward from $x=12$.

\[ \frac{ \qty{\hat \mu(12) - \hat\mu(11)} + \qty{ \hat\mu(11) - \hat\mu(10) } + \qty{ \hat \mu(10) - \hat\mu(9) } + \qty{ \hat \mu(9) - \hat\mu(8) } }{4} = \frac{\hat\mu(12) - \hat\mu(8)}{4} \]

As we go from $x=8$ to $x=12$, the ‘rise’ is $\hat\mu(12)-\hat\mu(8)$ and the ‘run’ is $12-8=4$; slope= rise / run.

Different Ways of Aggregating

The average, over people with ≥1 year of high school, of the increments affecting them in their last year.
- That’s something different. For the majority of these people, their last year was their fourth.
- So most of the terms we’re averaging are the increment from the third to fourth year.

\[ \frac{\sum_{i:X_i \in 9 \ldots 12} \qty{ \hat\mu(X_i) - \hat\mu(X_i-1) }}{\sum_{i:X_i \in 9 \ldots 12} 1} = \sum_{x \in 9 \ldots 12} P_x \ \qty{ \hat\mu(x) - \hat\mu(x-1) } \quad \text{ for } \quad P_x = \frac{N_x}{\sum_{x \in 9 \ldots 12}N_x}. \]

We can try the ‘cancellation trick’ we used before, but it doesn’t work very well.

\[\small{ \begin{aligned} & P_{12} \qty{ \hat\mu(12) - \hat \mu(11) } + P_{11} \qty{ \hat\mu(11) - \hat \mu(10) } + P_{10} \qty{ \hat\mu(10) - \hat \mu(9) } + P_{9} \qty{ \hat\mu(9) - \hat \mu(8) } \\ &= P_{12}\hat\mu(12) + (P_{11}-P_{12}) \hat\mu(11) + (P_{10}-P_{11}) \hat\mu(10) + (P_9 - P_{10}) \hat \mu(9) - P_9\hat \mu(8) \end{aligned}} \]

We do get a linear combination of the fine-group means, but it’s not an interpretable one.
It will, however, be useful in calculations next time when we’re talking about inference.

Different Results

\[ \small{ \begin{array}{r|rrrr} \text{Grade Increment} & 8 \to 9 & 9 \to 10 & 10 \to 11 & 11 \to 12 \\ \text{Mean Income Increment} & 1.64K & -12.64K & 11.96K & 8.52K \\ \text{Proportion of People} & 0.04 & 0.02 & 0.10 & 0.84 \\ \end{array} } \]

Question. Which is bigger—the average over years or the average over people?

Answer. The average over people is bigger. 2.37K vs 8.16K.

Our Weights

We can see the weights we’re using in the two summaries we’ve been talking about.
If you look at them hard enough, there’s a lot to see. But we won’t go into it here.
What’s obvious is that they focus, to different degrees, on different comparisons.
- The average over years puts all its emphasis on a long-term comparison: $\hat\mu(12)$ vs. $\hat\mu(8)$.
- The average over people puts most of its emphasis on a short-term one: $\hat\mu(12)$ vs. $\hat\mu(11)$.

The Impact of Variation in Slope

While our two summaries estimates are based on very different comparisons, we got similar answers.
That was, fundamentally, because every comparison tells us the same thing.
- If you draw the lines from $\hat\mu(8) \ldots \hat \mu(11)$ to $\hat\mu(12)$, you’d get roughly the same slope.
If that weren’t the case, it’d be a very different story. It’d look more like this.
In short, sometimes the choice of summary — among qualitatively similar ones — matters a lot.
Sometimes it doesn’t. But that’s not something you should bet much on.
We’ll talk more about choosing the right summary in the coming weeks.
For now, let’s dig into this phenomenon. We’ll look at data corresponding to a few stories about what’s useful.

Three Stories About High School

All that matters is the diploma.
All that matters is 10th grade algebra.
They teach the most important stuff first.

\[ \small{ \begin{array}{r|rrr} \text{Story} & 1 & 2 & 3 \\ \hline \text{Average over People} & \qquad & \qquad & \qquad \\ \text{Average over Years} & \qquad & \qquad & \qquad \\ \end{array} } \]

\[ \small{ \begin{array}{r|rrr} \text{Story} & 1 & 2 & 3 \\ \hline \text{Average over People} & 8.47K & 736.20 & 931.81 \\ \text{Average over Years} & 2.50K & 2.50K & 2.50K \\ \end{array} } \]

Which is Which?

Summary

Working with non-binary covariates, e.g., years of schooling, opens up a lot of possibilities.
- We can make and report comparisons on the finest level available, e.g. grade 12 vs grade 11.
- We can work with coarser groups, e.g. grade 12 vs grade < 12, to simplify what we have to report.
- We can make a number of fine comparisons and aggregate them into a simple summary.
For the most part, ultimately we’ll be looking at linear combinations of the subsample means.
This will be convenient next time when we’re seeing what the sample summaries tell us about our population.
It means that, in abstract terms, we’ll only need to understand the statistical behavior of one thing.